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Integral bounds

  1. Dec 3, 2014 #1
    1. The problem statement, all variables and given/known data
    I am having trouble setting up the bounds on the following two integrals:

    (a) The region E bounded by the paraboloid y=x2+z2 and the plane y=4.
    (b) The region bounded by the cylinder x2+y2=1, z=4, and the paraboloid z=1-x2-y2.

    2. Relevant equations

    3. The attempt at a solution
    I thought for (a) to use 4 < y < x2+z2
    -sqrt(y-z2) < x < sqrt(y-z2)
    -sqrt(y-x2) < z < sqrt(y-x2)
    But these don't seem right.
    I'm not sure where to begin for (b) except that z may be upper bounded by 4?
    Thanks in advance.
  2. jcsd
  3. Dec 3, 2014 #2


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    (a) The conditions you put on y will not give you a bounded region. For simplicity, I suggest you start working with ##r=\sqrt{x^2 + z^2}## instead of x and z.

    (b) As in (a), polar coordinates will serve you well here.
  4. Dec 3, 2014 #3


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    For these you want to use cylindrical coordinates. For (a), you should take [itex](x,y,z) = (r \cos \theta, y, r \sin \theta)[/itex]. For (b), you should take [itex](x,y,z) = (r \cos\theta, r \sin \theta, z)[/itex].
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